Theorem leeq1d 37475

Description: Specialization of breq1d 4593 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypotheses

Ref	Expression
leeq1d.1	⊢ (𝜑 → 𝐴 ≤ 𝐶)
leeq1d.2	⊢ (𝜑 → 𝐴 = 𝐵)
leeq1d.3	⊢ (𝜑 → 𝐴 ∈ ℝ)
leeq1d.4	⊢ (𝜑 → 𝐶 ∈ ℝ)

Assertion

Ref	Expression
leeq1d	⊢ (𝜑 → 𝐵 ≤ 𝐶)

Proof of Theorem leeq1d

Step	Hyp	Ref	Expression
1		leeq1d.1	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
2		leeq1d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	breq1d 4593	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐶 ↔ 𝐵 ≤ 𝐶))
4	1, 3	mpbid 221	1 ⊢ (𝜑 → 𝐵 ≤ 𝐶)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ℝcr 9814   ≤ cle 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584

This theorem is referenced by:  imo72b2lem0  37487